Journal of Statistical Physics manuscript No.(will be inserted by the editor)

Aaron Spettl · Thomas Werz ·Carl E. Krill III · Volker Schmidt

Parametric representation of 3D grainensembles in polycrystalline microstructures

Received: date / Accepted: date

Abstract As a straightforward generalization of the well-known Voronoi con-struction, Laguerre tessellations have long found application in the modelling,analysis and simulation of polycrystalline microstructures. The application of La-guerre tessellations to real (as opposed to computed) microstructures—such asthose obtained by modern 3D characterization techniques like x-ray microtomog-raphy or focused-ion-beam serial sectioning—is hindered by the mathematicaldifficulty of determining the correct seed location and weighting factor for each ofthe grains in the measured volume. In this paper, we propose an alternative to theLaguerre approach, representing grain ensembles with convex cells parametrizedby orthogonal regression with respect to 3D image data. Applying our algorithmto artificial microstructures and to microtomographic datasets of an Al-5 wt% Cualloy, we demonstrate that the new approach represents statistical features of theunderlying data—like distributions of grain sizes and coordination numbers—aswell as or better than a recently introduced approximation method based on theLaguerre tessellation; furthermore, our method reproduces the local arrangementof grains (i.e., grain shapes and connectivities) much more accurately. The addi-tional computational cost associated with orthogonal regression is marginal.

Keywords polycrystalline microstructure· Laguerre tessellation· orthogonalregression· convex cells

A. Spettl· V. SchmidtInstitute of Stochastics, Ulm University, Helmholtzstr. 18, 89069 Ulm, GermanyTel.: +49-731-50-23555Fax: +49-731-50-23649E-mail: aaron.spettl@uni-ulm.de

T. Werz· C. E. Krill IIIInstitute of Micro and Nanomaterials, Ulm University, Albert-Einstein-Allee 47, 89081 Ulm,Germany

2

1 Introduction

Inspired by experience gained from working with metals, mankind has long em-ployed various processing steps—such as mechanical deformation or heat treatment—to improve the properties of crystalline solids. It wasn’t until the past 100 years,however, that materials scientists were able to place this empirical knowledge on afirm scientific footing, attributing processing-induced property changes to the gen-eration and annihilation of lattice defects (grain boundaries, dislocations, pointdefects,etc.) constituting the microstructure of crystalline materials. And onlyduring the past decade has a determination of the true three-dimensional arrange-ment of various microstructural elements—such as the network of grain bound-aries spanning a polycrystalline specimen—become routinely feasible, thanks topowerful characterization techniques like x-ray microtomography [19,1], focused-ion-beam-based serial sectioning [7,25] and diffraction-based x-ray microscopy[21,17].

The increasing utilization of these new methods to map out polycrystallinemicrostructures in 3D has spurred newfound interest in developing mathematicalmodels for space-filling grain ensembles—i.e., tessellations of space—with em-phasis placed on matching not only the statistically averaged grain morphology inreal microstructures, but also their grain connectivitiesand local environments. Apopular starting point for such modelling efforts is the Laguerre tessellation (see[15] or Appendix A of the present paper), which is a generalization of the Voronoiconstruction, with weighting factors assigned to “seed points” offering rather flex-ible control of the resulting cell sizes. As with the conventional Voronoi diagram,the Laguerre tessellation consists of a set of non-overlapping convex cells that fillspace.

Because Laguerre tessellations are simple to define and construct mathemati-cally, they are attractive for the modelling of grain ensembles and their dynamics.For example, static polycrystalline structures have been represented by Laguerretessellations based on close-packed spheres with randomlyassigned radii [3], andthe coarsening of grains has been modelled and simulated using 2D and 3D La-guerre tessellations [24,22,23,28]. Furthermore, Laguerre tessellations have beenapplied to the modelling of other materials, like open and closed-cell foams [11,10]. Interestingly, it can be proven that every normal tessellation in 3D that con-sists entirely of convex cells is a Laguerre tessellation [9]. A normal tessellationis one in which adjacent cells are juxtaposed face-to-face,sharing not only a face,but also its edges and vertices; furthermore, each planar face is contained in ex-actly two cells, each edge borders exactly three cells, and each vertex is shared byexactly four cells. These conditions are typically satisfied by real polycrystallinemicrostructures, as well, despite the fact that most grainsare slightly non-convexthanks to the presence of curved boundaries. If we approximate the latter by pla-nar surfaces bounded by lines connecting the correspondinggrain vertices, thenwe can map any such real microstructure to a convex tessellation and, therefore,to a Laguerre tessellation.

Although it is straightforward to compare the statistical properties of cell sizes,shapes and environments in a given Laguerre tessellation tothe same propertiesevaluated for the grains of a real polycrystalline specimen, it is no easy task todetermine the Laguerre tessellation that corresponds mostclosely to a particular

3

grain mapping. This would entail finding the set of seed points and weighting fac-tors that, when inserted into the Laguerre construction, leads to a network of grainboundaries best matching the one measured experimentally.But in a Laguerre tes-sellation, there is no guarantee that the seed point for a given cell actually lieswithin that cell, as the seed point location depends strongly on the positions andweighting factors of all nearby seed points, which are also unknown. To solvethis optimization problem, one could, for example, devise astrategy for identi-fying and iteratively varying the seed point locations and weighting factors untila predetermined convergence criterion is fulfilled. We refer to the output of suchan approach as aLaguerre approximationto the experimental microstructure. Aprescription for constructing a Laguerre approximation was formulated recentlyby Lyckegaardet al. [12], who proposed starting from the centroids and volume-equivalent radii of individual grains and then applying an additional optimizationstep. The resulting representation faithfully reproducesthe statistical propertiesof the experimental microstructure, but, according to the authors of [12], theirLaguerre approximation is not always able to capture local grain configurationsprecisely. While this method aims primarily at reconstructing the full morphol-ogy when only grain centres and grain volumes are known, it isalso applicable tovoxelated data.

Lacking a satisfactory solution to this complex optimization problem, we pro-pose an alternative route to the extraction of a set of convexpolyhedra that closelymatches the grain mapping of any given real polycrystal. Ourapproach offersmost of the advantages of the Laguerre tessellation withoutthe burden of havingto find seed points and weighting factors for the individual cells. In our method,grains are described parametrically, with the individual faces obtained by orthogo-nal regression of planes; subsequently, the cells are constructed by an appropriatecombination of the planar faces. Just like the Laguerre tessellation, our parametricrepresentation of the grains affords significant data reduction, while facilitatingthe estimation of structural characteristics for individual grains. The proposed al-gorithm is explained in Section 2. We test the algorithm by applying it to artificial3D image data and to a real microtomography data set obtainedfrom an Al-5wt% Cu alloy, comparing the algorithm’s performance to the Laguerre approxi-mation devised by Lyckegaardet al. [12] (Section 3). We observe that both ourapproach and the Laguerre approximation successfully reproduce statistical fea-tures of the real ensemble of grains, such as the distributions of grain sizes andcoordination numbers, but our algorithm delivers a significantly closer match tolocal grain properties, such as the grain size, shape and neighbourhoods, at thecost of a slightly more complex parametrization.

2 Representation of grains by parametric cells

In this section, we describe a new algorithm for extracting convex cells directlyfrom 3D image data. It is assumed that the voxels belonging toeach grain are la-belled with a unique number—that is, the processed imageI is given by{I(x,y,z)∈{1, . . . ,N} : (x,y,z)∈W}, whereN ≥ 1 denotes the number of grains andW ⊂N

3

is the (convex) grid of voxel coordinates;N = {0,1, . . .}. Furthermore, we as-sume that all voxels belonging to a given grain with labeli ∈ {1, . . . ,N}—i.e.,Ri = {(x,y,z) ∈W : I(x,y,z) = i}—form a connected region.

4

A convex cell can be described by an intersection of half-spaces, in which theboundary of each half-space is aligned along a single planarface of the cell. Bylooking at the individual faces of a given grain—that is, thevoxels bordering agrain and one of its neighbours—we can fit a plane to those voxels using orthog-onal regression and then define a closed cell from the set of all planes delimitingthe given grain. A 2D illustration of this idea is shown in Figure 1(a).

(a) (b)

Fig. 1 Schematic illustration of the cell extraction algorithm in2D: (a) orthogonal regression isused to fit lines to pixels at each boundary between two grains; taken together, the set of linessurrounding the grain defines the extracted cell. (b) Implementation of this idea in 2D usinghalf-planes: orthogonal regression is used to fit lines to pixels (shown here only for two grainboundaries), the corresponding half-planes are computed (grey), and the intersection of all half-spaces (dark grey) is taken to define the cell. The filled circle denotes the grain’s centroid, whichmust be located within the half-planes.

A prerequisite for this approach to succeed is that the grains under consider-ation be at least nearly convex. For convex grains having planar faces, the algo-rithm works perfectly, in the sense that the precision is limited only by the voxelresolution. Non-planar faces between grains are approximated by planar faces; thepresence of non-convex grains can introduce artefacts, as discussed in Section 3.3.

We now describe the extraction of planes for individual grain faces and theconstruction of cells from sets of planar faces.

2.1 Extraction of a plane for an individual grain face

In a typical polycrystalline microstructure, the voxels insimultaneous contact withtwo grainsi and j are located approximately along a plane in 3D. We wish to detectthis plane using orthogonal regression. The planePn,d ⊂ R

3 parametrized with a(unit-length) normal vectorn = (n1,n2,n3) ∈ R

3 and signed distanced ∈ R fromthe origin is given by

Pn,d = {(x,y,z) ∈ R3 : n1x+n2y+n3z+d = 0} .

The set of voxels adjacent to two grains with labelsi and j is

Ni, j = {(x,y,z) ∈W : N26(x,y,z)∩Ri 6= /0 andN26(x,y,z)∩Rj 6= /0} ,

5

whereN26(x,y,z) denotes the set of 26 voxels that share a face, edge or cornerwith the voxel at(x,y,z) (i.e., the voxel neighbourhood defined by a distance lessthan or equal to

√3 from (x,y,z)). Note thatNi, j = Nj,i , which ensures that the

detected planes do not depend on the order of processing.The idea of orthogonal regression is to minimize the (squared) distance of

points (in our case the voxel coordinates inNi, j ) from the plane that is to bedetected. Formally, we want to find the (global) minimum of the function f :R

3×R 7→ [0,∞) denoted by

f (n,d) = ∑(x,y,z)∈Ni, j

(n1x+n2y+n3z+d)2

for a normal vectorn = (n1,n2,n3) ∈ R3 with length|n|= 1 and signed distance

d ∈R from the origin. Orthogonal regression is therefore a least-squares problem,which can be solved by singular value decomposition [4]. In essence, for a givenset of boundary voxels, it can be shown that the centroidc∈ R

3 of the voxels lieson the plane to be extracted, and a normal vectorn ∈ R

3 to the plane is given bythe right-singular vector corresponding to the smallest singular value of a matrixcontaining all voxel coordinates shifted by−c [4].

2.2 Determination of grains sharing a common face

We first consider the issue of deciding which pairs of grainsRi andRj share acommon face, since this determines whether a plane between the grains shouldbe extracted or not. Note that the ruleNi, j 6= /0 does not suffice for this purpose,because there may be voxels touching two grains that share only a common vertexor edge. Fitting a plane to these voxels has no physical justification and wouldlikely generate odd and unpredictable cell shapes. To avoidthis situation, we applya threshold criterion, with the value of the threshold determined from an inspectionof the data in question. For the sample data sets discussed inSection 3, we interpretthe square root of the number of voxels inNi, j as an “effective face diameter,”under the provisional assumption that the voxels inNi, j arise primarily from ashared grain face. We then calculate the ratio between the effective face diameterand the mean volume-equivalent diameter of the two grains being evaluated. Ifthis ratio is larger than the threshold value—typically in the range of 0.2 to 0.25—then we accept the voxels inNi, j as belonging to a grain face, and we apply thealgorithm of Section 2.1 to extract the corresponding best-fit plane.

2.3 Cell representation by the intersection of half-spaces

As explained above, we can fit a plane to each face of a grainRi , and each of theseplanes restricts the volume of the corresponding extractedcell; consequently, thecell is a convex polytope. Convex polytopes can be defined by their vertices (thepolytope itself is then given by the convex hull of the vertices, sometimes calledthe V-representation) or by an intersection of half-spaces(H-representation). Thelatter is directly applicable to our approach, because the appropriate half-spacesare directly related to the detected planes.

6

A half-spaceHn,d in R3 is given by

Hn,d = {(x,y,z) ∈ R3 : n1x+n2y+n3z+d ≤ 0} , (⋆)

wheren = (n1,n2,n3) ∈ R3 is a (unit-length) normal vector, andd ∈ R is the

signed distance from the origin. It is straightforward to obtain these half-spacesfrom the previously detected planesPn,d, which are the boundaries of the half-spaces. Of course the proper side for the half-space must also be determined; here,a simple criterion is to require the half-space in(⋆) to include the centroid ofRi—i.e., the grain centroid fulfils the inequality given in the definition of Hn,d.

Finally, following the steps previously described, we can extract planes for allfaces of a given grain, determine the corresponding half-spaces and obtain a closedcell by taking their intersection. A schematic illustration is shown in Figure 1(b).For m half-spaces of a single grain withm≥ 3 and{(n(k),d(k)) ∈ R

3 ×R,k =1, . . . ,m}, their intersection and thus the convex cellC is given by

C= {(x,y,z) ∈ R3 : n(k)1 x+n(k)2 y+n(k)3 z+d(k) ≤ 0, k= 1, . . . ,m} .

3 Evaluation

We evaluate the quality of the proposed cell extraction algorithm by computingstructural characteristics. First, using the standard Laguerre construction (see Ap-pendix A), we generate an artificial data set consisting of a space-filling ensembleof convex cells (taking on the role of the grains in a polycrystalline microstruc-ture), to which we apply our algorithm in order to test the accuracy of the resultingcell extraction. Second, we apply our algorithm to the grains of a real polycrys-talline material mapped experimentally by x-ray microtomography.

In both cases, we compare our approach to the Laguerre approximation in-troduced by Lyckegaardet al. [12]. In the latter, seed points are placed initiallyat the centroids of individual grains, and a weighting factor equal to the volume-equivalent radius of the associated grain is assigned to each seed point. An addi-tional optimization step is carried out to improve the result by shifting the locationof seed points depending on adjacent cells. From the analysis carried out in [12],Lyckegaardet al. conclude that this cell-extraction algorithm reproduces statisti-cal features like the distribution of cell sizes and the distribution of the number offaces quite well, but discrepancies can appear in more-sophisticated local charac-teristics, such as the sizes and shapes of cell neighbourhoods.

Throughout Section 3, when estimating structural characteristics we employthe Miles–Lantuejoul correction [14,8] to avoid bias arising from grains touchingthe boundary of the grid, and, whenever possible, we prefer to calculate densityfunctions rather than histograms, as the shape of the latteris highly sensitive tothe chosen bin widths and positions. In this work, we determine density functionsby (non-parametric) kernel density estimation [20].

3.1 Artificial data set with convex cells

Here, we describe the construction of an artificial polycrystalline microstructure asa realization of a random Laguerre tessellation. We then apply our cell-extraction

7

algorithm and compare the resulting cells to those of the original known tessella-tion.

3.1.1 Generation of the artificial data set

A random Laguerre tessellation is generated by consideringa random markedpoint process [6], the points of which are taken to be seed points for Laguerrecells. Numerical marks are assigned randomly to the points,constituting weight-ing factors that influence the resulting sizes of the Laguerre cells. More precisely,we employ a Poisson-Laguerre tessellation [9], which is based on the well-known(homogeneous) Poisson point process. In particular, the point pattern is gener-ated as follows. For a given intensityλ of the homogeneous Poisson process,the number of pointsN in a cuboidW ⊂ R

3 is Poisson-distributed with expecta-tion value given by the product ofλ and the volume ofW. For a realizationn ofthe number of points, the point pattern itself is denoted by the set of coordinates{(xi ,yi ,zi), i = 1, . . . ,n}; these points are realized by a uniform distribution onW,and they do not influence each other.

We choose the cuboid for simulation to have the sizeW = [0,1000]3 andλ tobe 5×10−6, which means that the expected number of points inW is 5000. Therandom marks assigned to the points are independent and distributed according toa gamma distribution with shape parameterα = 50 and rate parameterβ = 0.75(expectation valueα/β and varianceα/β2) [5]. This combination of parametervalues was found to generate cells similar to the experimental data presented inSection 3.2. A 2D section of a realization of this random Laguerre tessellation isshown in Figure 2.

Fig. 2 Cross-section through a 3D artificial data set generated by Poisson-Laguerre tessellation.

3.1.2 Extraction of cells and evaluation

Given that the Poisson-Laguerre tessellation described above has been discretizedto a cubic grid, it is straightforward to apply the cell-extraction algorithm proposed

8

in Section 2. For the sake of comparison, we also compute the Laguerre approx-imation of Lyckegaardet al. [12] for the same artificial data set. Figure 3 showsthe original tessellation overlaid with cells extracted bythe two methods. To quan-tify the degree of agreement between the extracted and original cells, we comparethem at the voxel level. In light of the near-perfect match between the boundaryvoxels extracted by our approach and those in the original data set, it is clear thatall structural characteristics, like cell sizes, cell faces and cell neighbourhoods, arealso identical.

(a) (b)

Fig. 3 Cross-section through the original Poisson-Laguerre tessellation, with boundary voxelsshaded grey. Superimposed in black are the boundaries of cells extracted by (a) orthogonalregression and (b) Laguerre approximation [12].

In order to quantify the number of voxels correctly assigned, we examine theset of voxels of an extracted cellC, which should be nearly identical to that ofthe corresponding original cellC. We define the fraction of correct voxels as thenumber of voxels in the intersectionC∩C divided by the number of voxels inC.The fraction of correct voxels is a number lying between 0 and1, with values nearunity implying a nearly perfect match to the original voxels. Note that an extractedcell that happens to be larger than the original cell can alsoyield a high value forthis fraction, even though the extracted cell is not a perfect fit, but in this case theadjacent cells will automatically manifest much lower values for the fraction ofcorrect voxels.

Figure 4 shows the estimated density function of the fraction of correct voxels.For cells constructed by orthogonal regression, the mean fraction value is about99%, which constitutes a very good overall fit. This is not surprising, becauseour algorithm is constructed to match the individual faces as closely as possiblewith planes, and the artificial data set consists entirely ofconvex cells havingplanar faces. The Laguerre approximation [12], on the otherhand, is generatedby a simpler parametrization. Because the original data setis itself a Laguerretessellation, it is clear that the cells extracted by the algorithm in [12] could haveconceivably matched perfectly throughout the sample, but the fact that the meanfraction of correct voxels was only 84% for the Laguerre approximation indicates

9

that determination of the proper seed point locations and weighting factors fromcell boundaries alone is a non-trivial endeavour.

The voxel-based comparisons presented in Figures 3 and 4 indicate that for(in our sense) optimal data, the extraction of cells by orthogonal regression worksnearly perfectly. Despite the artificial data set having been generated by the La-guerre construction, the Laguerre approximation [12] doesnot describe the datanearly as well. For the experimental data set considered in the next section, weextend our analysis beyond the voxel level to include characteristics like the sizesand shapes of grains and their neighbourhoods.

0

5

10

15

20

25

30

0.75 0.8 0.85 0.9 0.95 1

dens

ity

fraction of correct voxels

orthogonal regressionLaguerre approximation

Fig. 4 Estimated density function plotted against the fraction ofvoxels correctly assigned in theartificial data set for cells extracted by orthogonal regression and by Laguerre approximation[12].

3.2 Experimental data set of a polycrystalline alloy

After a short description of the method by which 3D microstructural data wereobtained for a polycrystalline alloy, we examine the cells determined by orthog-onal regression and present various measures for quantifying the extent to whichthe extracted cells properly represent the real microstructure.

3.2.1 Data description

Cylindrical samples of an Al-5 wt% Cu alloy (4 mm diameter, 4 mm height) werecharacterized using a SkyScan 1172 laboratory tomograph atan isotropic resolu-tion of 2 µm (voxel side length). Absorption-contrast microtomography yields athree-dimensional mapping of the local absorption of x-rays, variations in whichcan arise, for instance, from concentration gradients present at the boundaries be-tween different phases. When Al-Cu alloys with low copper content are heatedabove the solidus temperature, phase separation occurs into solid, Al-rich grainssurrounded by a liquid phase having a significantly higher concentration of Cuatoms [13,16]. Upon subsequent cooling to room temperature, the liquid phasecrystallizes into an eutectic mixture of two solid phases, but the local Cu concen-tration in the formerly liquid regions remains enhanced, effectively marking theboundaries of the Al-rich grains in the tomographic reconstruction (Figure 5(a)).Because coverage of the grains by the eutectic mixture was incomplete, we applied

10

a watershed image-processing algorithm to a Euclidean distance transform of thebinary images to fill in gaps in the network of grain boundaries (Figure 5(b)) [2,18]. Additional details concerning the measurement and data segmentation proce-dures can be found in [27].

(a) (b)

Fig. 5 (a) Cross-section through a tomographic reconstruction ofan Al-5 wt% Cu alloy mea-sured after cooling from above the solidus down to room temperature; lighter regions indicate thepresence of the phase Al2Cu, which forms during crystallization of the liquid phase surroundingthe solid, Al-rich grains. (b) Same cross-section following binarization and application of thewatershed transformation.

3.2.2 Extraction of cells and evaluation

As mentioned above, the watershed transformation was used to reconstruct thenetwork of grain boundaries, thereby uniquely identifyingconnected sets of voxelsas belonging to the interior regions of individual grains inthe sample. This isprecisely the starting point considered in Section 2 for theextraction of cells froma 3D mapping of a polycrystalline microstructure. Application of this method tothe Al-5 wt% Cu data set yields the cell system illustrated inFigure 6(a) in black,which is superimposed on the boundary network of the original tomographic dataplotted in grey.

To assess the accuracy of cell extraction, we again considerthe fraction of cor-rect voxels, as defined in Section 3.1; the estimated densityfunction is plotted inFigure 7. As with the artificial data set, we see that orthogonal regression appliedto data from a real sample yields very high values for the meanfraction of cor-rect voxels (0.96), with only 5% of all cells having a value smaller than 0.9. TheLaguerre approximation [12] achieves a mean fraction of about 0.87, but 5% ofthe cells manifest a value below 0.64. This quantitative discrepancy is reflected inobvious qualitative differences between the overlays shown in Figure 6.

3.2.3 Comparison of further structural characteristics

In this section, we consider important structural characteristics of polycrystallinemicrostructures with respect to grain sizes, shapes and neighbourhoods, compar-

11

(a) (b)

Fig. 6 Cross-section through a tomographic reconstruction of Al-5 wt% Cu following imageprocessing as described in the text; voxels at grain boundaries are shaded grey. Superimposedin black are the boundaries of cells extracted by (a) orthogonal regression and (b) Laguerreapproximation [12].

0

5

10

15

20

25

0.5 0.6 0.7 0.8 0.9 1

dens

ity

fraction of correct voxels

orthogonal regressionLaguerre approximation

Fig. 7 Estimated density function plotted against the fraction ofvoxels correctly assigned in theAl-5 wt% Cu data set for cells extracted by orthogonal regression and by Laguerre approxima-tion [12].

ing the corresponding values for cells extracted by orthogonal regression and byLaguerre approximation of the same experimental Al-5 wt% Cudata set.

A natural structural characteristic is the distribution ofgrain sizes. Althoughthe grains themselves are not perfectly spherical, it is common to equate the sizeof a given grain to the diameter of a sphere having the same volume. The esti-mated density functions for grain sizes computed in this manner are plotted inFigure 8(a). We see there that all of the distributions take on a similar shape, butsome discrepancies are apparent between the experimental data set and the La-guerre approximation, particularly with respect to smaller grains, which tend to berepresented by Laguerre cells that are somewhat too large. The near-linearity of ascatter plot of the diameter of each extracted cell against its corresponding tomo-graphic grain diameter attests to the high accuracy our cellextraction algorithm(Figure 8(b)). The few outliers can be attributed to the presence of non-convexcells in the real microstructure (see Section 3.3).

Another interesting structural characteristic is the sphericity of grains [26].Sphericityis defined to be the ratio of the surface area of the volume-equivalent

12

(a)

0

0.001

0.002

0.003

0.004

0.005

0 100 200 300 400 500 600 700 800

dens

ity

volume-equivalent diameter (µm)

tomographic dataorthogonal regression

Laguerre approximation

(b)

0

100

200

300

400

500

600

700

0 100 200 300 400 500 600 700

diam

eter

s in

ext

ract

ed c

ells

(µm

)

volume-equivalent diameters in tomographic data (µm)

orthogonal regressionLaguerre approximation

Fig. 8 (a) Estimated density functions for the volume-equivalentgrain diameter: distributionscalculated directly from tomographic data as well as from cells extracted by orthogonal re-gression or Laguerre approximation [12]. (b) Scatter plot of the volume-equivalent diameter ofextracted cellsvs.the corresponding grain diameter measured by tomography.

sphere to the surface area of the corresponding grain or cell. Small values for thisratio imply that the shape differs significantly from that ofa sphere—e.g.the trueshape could be elongated in a particular direction. As shownin Figure 9, neitherorthogonal regression nor the Laguerre approximation [12]does a satisfactory jobof representing the sphericity of the experimental grains:the mean sphericity isapproximately equal in all three cases, but the distribution of sphericities is widerfor both types of extracted cells than for the original grains. Detailed investiga-tion reveals that discrepancies are particularly noticeable for small grains, whichtend to be rather spherical in the experimental data set. Consequently, the approx-imation of planar faces (which is enforced by the cell extraction algorithms) canbe rather poor for these grains (particularly in light of thefact that small grainshave few faces), and the sphericity of the cells extracted for these grains neverapproaches unity. Therefore, the differences in sphericity that are clearly visiblein Figure 9 can be attributed primarily to the constraint of convexity on each cellin the extracted cell network (which, in turn, entails that all cell boundaries beplanar).

A further important characteristic of space-filling ensembles of grains is thecoordination number, which denotes the number of neighbourgrains sharing acommon face with a given grain. Figure 10(a) compares the relative frequencyof coordination numbers determined directly from experimental data to the samequantity evaluated for microstructural representations produced by the two cell-extraction algorithms. For the tomographic data set, the identification of sharedgrain faces was performed using the algorithm described in Section 2.2.

When assessing grain adjacency, it is of interest to determine whether thetopology of the system is represented properly. Figure 10(b) presents histogramsfor the number of incorrectly assigned neighbours, which isdetermined by count-ing all differences in the adjacency list—i.e., by summing up the number of miss-ing neighbours and the number of additional neighbours. As already discussed in[12], the cells extracted by Laguerre approximation are notperfect with respectto the local arrangement of cells. Cell extraction by orthogonal regression mani-fests similar deficiencies in the proper representation of this particular structuralcharacteristic, although they are somewhat less severe than in the case of Laguerreapproximation.

13

(a)

0

2

4

6

8

10

12

14

16

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

dens

ity

sphericity 0 100 200 300 400 500

diameter (µm)

0.7

0.75

0.8

0.85

0.9

0.95

1

sphe

ricity

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

frequency(b)

0

2

4

6

8

10

12

14

16

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

dens

ity

sphericity 0 100 200 300 400 500

diameter (µm)

0.7

0.75

0.8

0.85

0.9

0.95

1

sphe

ricity

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07frequency

(c)

0

2

4

6

8

10

12

14

16

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

dens

ity

sphericity 0 100 200 300 400 500

diameter (µm)

0.7

0.75

0.8

0.85

0.9

0.95

1

sphe

ricity

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

frequency

Fig. 9 Grain sphericity: (a) tomographic data acquired from Al-5 wt% Cu; (b) cells extractedby orthogonal regression; (c) cells extracted by Laguerre approximation [12]. In each row, thedensity function for sphericity is plotted on the left, and the dependence of sphericity on thegrain/cell size is illustrated graphically on the right.

For both cell-extraction routines, erroneous neighbour assignments can betraced to the constraint of cell convexity. When the orthogonal regression of planesis applied to the boundaries of a non-convex grain, it is possible for a neighbourto be lost when the voxels of the corresponding shared face donot generate ahalf-space that restricts the volume of the resulting cell.Approaches based on La-guerre tessellations do not suffer from this particular problem, but there—as wehave seen above—it is much more difficult to obtain a good overall fit to the indi-vidual grain faces, which may, in turn, lead to incorrectly assigned neighbours. Inthe following section, we discuss in greater detail the cell-extraction artefacts thatcan arise from non-convex grains.

3.3 Influence of non-convexity

The proposed algorithm aims to obtain an exact representation of individual grains.For an ensemble of convex grains with planar faces, our cell-extraction method

14

(a)

0

0.02

0.04

0.06

0.08

0.1

0.12

0 5 10 15 20 25 30 35 40

rela

tive

freq

uenc

y

coordination number

tomographic dataorthogonal regression

Laguerre approximation

(b)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10

rela

tive

freq

uenc

y

number of incorrectly assigned neighbors

orthogonal regressionLaguerre approximation

Fig. 10 (a) Distribution of the coordination number of grains in tomographic image data as wellas of cells extracted by orthogonal regression and Laguerreapproximation [12]. (b) Distributionof the number of incorrectly assigned neighbours: comparison of cells extracted by orthogonalregression to Laguerre approximation, evaluated in both cases with respect to grain neighbourassignments determined from experimental data.

works perfectly. However, the orthogonal regression approach does not explic-itly enforce constraints like the face-to-face adjacency of cells or the matching ofedges and vertices; consequently, it is highly likely that the conditions for nor-mality of a tessellation will be violated to a certain extentwhen our algorithm isapplied to a microstructure containing non-convex grains.We now discuss twotypes of artefacts that can arise from non-convexity.

First, our algorithm works by fitting a plane to the boundary between twograins. This ensures that we obtain a good fit with respect to the position of theboundary itself, but the approach places no constraints on the locations of theedges and vertices delineating the shared face. Consider, for example, the 2D il-lustration in Figure 11(a). Grain number 1 shares a curved boundary with grain3 below, which is clearly non-convex. By fitting lines to thisboundary and tothe boundary between grains 1 and 2, we obtain the vertex labelled V. The sameprocedure applied to the boundaries of grain 2, however, yields a different ver-tex positionV ′ for the same line separating cells 1 and 2. Such non-matchingofvertices in 2D—or edges and vertices in 3D—will generally result in small gaps—i.e. regions not covered by any extracted cell—or even in small overlaps betweenneighbouring cells. Examination of our data found that approximately 97% of allvoxels were included within an extracted cell, but this number also encompassesartefacts of the second type, which we discuss next.

A second type of artefact occurs when a detected half-space cuts off too muchvolume from a cell. This can be a particularly serious problem when the segmen-tation of an experimental data set fails to detect a boundarybetween two grains,falsely grouping the voxels together as a single grain. The union of the two grainsis usually highly non-convex, frequently taking on a “dog-bone” shape. Then, be-cause each planar face is detected individually and used to construct the resultingcell by taking the intersection of the corresponding half-spaces, some half-spacesmay not restrict the volume of the cell at all, or, even worse,they may strongly af-fect the overall cell shape at the wrong location. This generally results in extractedcells that are significantly smaller than the correspondinggrains, as illustrated inFigure 11(b). (Such an obvious loss of volume between the original grain and theextracted cell could be a useful tool for automatically detecting underlying seg-mentation errors.)

15

(a) (b)

(c)

Fig. 11 Schematic illustration of artefacts caused by non-convex grains. (a) Edges and verticesof adjacent cells are not forced to be equal, which may resultin small gaps (unshaded whiteregion). (b) Two grains recognized as one in the segmentation cannot be represented properly bya convex cell (detected cell bordered in solid blue lines, restricting planes dashed, lost volumein white); (c) A planar face can be “cut off” from one grain when another face more stronglyrestricts the volume of the extracted cell (the face marked in blue has no affect on the spatialextent of cell 3); since the former face may still be relevantfor an adjacent cell (cell 2 in theillustrated example), a gap can form between the extracted cells.

The misclassification of two grains as a single grain is clearly a fault of seg-mentation and not of cell extraction. Nevertheless, the artefact of a half-spacehaving no effect or even the wrong effect on the cell shape canoccur under less-extreme circumstances, as well. The enforced approximation of a curved boundaryby a planar face can generate a half-space that “cuts off” other fitted planar facesin such a manner that a “cut off” plane disappears as a face forone cell, but thesame plane forms the boundary of an adjacent cell. Such a caseis illustrated inFigure 11(c). Here, the plane fitted to the boundary between grains 2 and 3 is onlyrelevant for cell 2, since, for cell 3, this plane is cut off bythe half-space gener-ated from the (non-planar) boundary between grains 1 and 3. This phenomenoncan account for instances in which cells extracted by our algorithm have differentneighbours than the corresponding grains in the experimental data set.

In spite of such artefacts, our strategy for extracting (parametric) cells per-forms quite well in general for experimental data corresponding to typical poly-crystalline microstructures. As expected, the high accuracy noted in voxel-basedcomparisons carries over to statistical characteristics of the cell ensembles, aswell. The Laguerre approximation [12] is likewise able to represent global sta-tistical features faithfully, but it is less suited to the proper description of moresophisticated structural characteristics of space-filling grain ensembles. It shouldbe noted that Laguerre tessellations have the clear advantage of, by definition, al-ways generating a tessellation—i.e., it is impossible for there to be gaps between

16

cells or for edges and vertices not to coincide. Unfortunately, it is these same con-straints that make it so difficult to determine the “best” seed points and weightingfactors for accurately approximating a measured polycrystalline microstructure bythe Laguerre construction.

4 Conclusions

We have presented a new algorithm for extracting parametriccells from 3D imagedata that is based on the orthogonal regression of individual grain faces. Appliedto an artificial data set consisting of convex grains, the cell-extraction algorithmgenerates a nearly perfect match, as quantified by a voxel-based comparison. Forexperimental 3D image data, the new algorithm also works quite well, performingsignificantly better in most respects than a recently proposed Laguerre approxi-mation algorithm [12]. The latter conclusion is based on evaluations of statisticalfeatures like distributions of grain size and the number of neighbours (coordi-nation number), but also on local characteristics like the number of incorrectlyassigned neighbours per grain. The higher accuracy in representation comes at thecost of a slightly more complex parametrization. It should be noted that voxelated(image) data are needed for our approach, while [12] requires only grain centresand volumes, which may be easier to obtain. A further disadvantage is the pos-sibility for artefacts like small gaps between adjacent cells in the case of clearlynon-convex grains. The latter problem was negligible for the real microstructurestudied in this work.

Acknowledgements The authors would like to thank D. Molodov of the Institute ofPhysi-cal Metallurgy and Metal Physics, RWTH Aachen, for sample preparation; the Institute of Or-thopaedic Research and Biomechanics, Ulm University, for x-ray microtomography beamtime;and especially Uwe Wolfram for assistance with the tomography measurements and numerousdiscussions. Furthermore, the authors are grateful to the Deutsche Forschungsgemeinschaft forfunding through NSF/DFG Materials World Network Project KR1658/4-1.

A Laguerre tessellations

The Laguerre tessellation [15]—a weighted version of the well-known Voronoi diagram—haslong been a popular choice for modelling polycrystalline grain structures (seee.g.[3,24,22,23,28]). In a Voronoi diagram, each “seed point” (also called a “spring point”) creates exactly onecell, as every point in space is assigned to its nearest seed point. In 3D, the result is a partitionof space into convex polyhedra. When constructing a Laguerre tessellation, every seed point isgiven an additional weightr2, which permits finer control over the cell sizes. The weightr2 canbe interpreted as the squared radius of a sphere centred on the seed point—see Figure 12 for anillustration in 2D.

Formally, a 3D Laguerre tessellation is defined as follows. Given a (locally finite) setS={(xi , r i), i ∈ I} ⊂R

3×R+ of seed pointsxi with radii r i , the Laguerre cell of(xi , r i) with respect

to S is given by

C((xi , r i),S) ={

y ∈ R3 : |y−xi |2− r2

i ≤∣∣y−x j

∣∣2− r2j , (x j , r j ) ∈ S

}.

Then, the Laguerre tessellation is the set of all Laguerre cells {C((xi , r i),S), i ∈ I}. Note that it ispossible for a seed point to create no cell at all, provided anadjacent seed point has a sufficientlylarge weight. For the same reason it is also possible for a seed point not to be contained within

17

Fig. 12 Illustration of Laguerre tessellation in 2D: seed points with weights pictured as circles,along with the resulting Laguerre cell boundaries.

the cell generated by that seed point. These properties complicate the determination of the seedpoint locations and corresponding weights when only the resulting cells are known.

References

1. Aagesen, L.K., Fife, J.L., Lauridsen, E.M., Voorhees, P.W.: The evolution of interfacialmorphology during coarsening: A comparison between 4D experiments and phase-fieldsimulations. Scr. Mater.64(5), 394–397 (2011)

2. Brunke, O., Odenbach, S., Beckmann, F.: Quantitative methods for the analysis ofsynchrotron-µCT datasets of metallic foams. Eur. Phys. J.-Appl. Phys.29(1), 73–81 (2005)

3. Fan, Z., Wu, Y., Zhao, X., Lu, Y.: Simulation of polycrystalline structure with Voronoidiagram in Laguerre geometry based on random closed packingof spheres. Comp. Mater.Sci.29, 301–308 (2004)

4. de Groen, P.P.N.: An introduction to total least squares.Nieuw Archief voor Wiskunde, 4thSeries14, 237–253 (1996)

5. Hogg, R.V., Craig, A.T.: Introduction to Mathematical Statistics, 4th edn. Macmillan, NewYork (1978)

6. Illian, J., Penttinen, A., Stoyan, H., Stoyan, D.: Statistical Analysis and Modelling of SpatialPoint Patterns. J. Wiley & Sons, Chichester (2008)

7. Konrad, J., Zaefferer, S., Raabe, D.: Investigation of orientation gradients around a hardLaves particle in a warm-rolled Fe3Al-based alloy using a 3DEBSD-FIB technique. ActaMater.54(5), 1369–1380 (2006)

8. Lantuejoul, C.: On the estimation of mean values in individual analysis of particles. Mi-crosc. Acta5, 266–273 (1980)

9. Lautensack, C.: Random Laguerre Tessellations. Ph.D. thesis, Universitat Karlsruhe, Weilerbei Bingen (2007)

10. Lautensack, C.: Fitting three-dimensional Laguerre tessellations to foam structures. J. Appl.Stat.35(9), 985–995 (2008)

11. Lautensack, C., Sych, T.: 3D image analysis of open foamsusing random tessellations.Image Anal. Stereol.25, 87–93 (2006)

12. Lyckegaard, A., Lauridsen, E.M., Ludwig, W., Fonda, R.W., Poulsen, H.F.: On the use ofLaguerre tessellations for representations of 3D grain structures. Adv. Eng. Mater.13(3),165–170 (2011)

13. Massalski, T.B.: Binary Alloy Phase Diagrams - AlCu-Phasediagram, vol. 1, 3rd edn. ASMInternational (1996)

14. Miles, R.E.: On the elimination of edge effects in planarsampling. In: E.F. Harding,D.G. Kendall (eds.) Stochastic Geometry, pp. 228–247. J. Wiley & Sons, New York (1974)

15. Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial Tessellations: Concepts and Appli-cations of Voronoi Diagrams, 2nd edn. J. Wiley & Sons, Chichester (2000)

18

16. Pompe, O., Rettenmayr, M.: Microstructural changes during quenching. J. Cryst. Growth192(1-2), 300–306 (1998)

17. Reischig, P., King, A., Nervo, L., Vigano, N., Guilhem,Y., Palenstijn, W.J., Batenburg, K.J.,Preuss, M., Ludwig, W.: Advances in X-ray diffraction contrast tomography: flexibility inthe setup geometry and application to multiphase materials. J. Appl. Crystallogr.46(2),297–311 (2013)

18. Roerdink, J.B.T.M., Meijster, A.: The watershed transform: definitions, algorithms, and par-allellization strategies. Fundam. Inform.41, 187–228 (2001)

19. Salvo, L., Cloetens, P., Maire, E., Zabler, S., Blandin,J.J., Buffiere, J.Y., Ludwig, W., Boller,E., Bellet, D., Josserond, C.: X-ray micro-tomography an attractive characterisation tech-nique in materials science. Nucl. Instrum. Methods Phys. Res. B200, 273–286 (2003)

20. Silverman, B.W.: Density Estimation for Statistics andData Analysis. Chapman &Hall/CRC, London (1986)

21. Sorensen, H.O., Schmidt, S., Wright, J.P., Vaughan, G.B.M., Techert, S., Garman, E.F.,Oddershede, J., Davaasambuu, J., Paithankar, K.S., Gundlach, C., Poulsen, H.F.: Multigraincrystallography. Z. Kristallogr.227(1), 63–78 (2012)

22. Telley, H., Liebling, T.M., Mocellin, A.: The Laguerre model of grain growth in two di-mensions: I. Cellular structures viewed as dynamical Laguerre tessellations. Philos. Mag.B 73(3), 395–408 (1996)

23. Telley, H., Liebling, T.M., Mocellin, A.: The Laguerre model of grain growth in two dimen-sions: II. Examples of coarsening simulations. Philos. Mag. B 73(3), 409–427 (1996)

24. Telley, H., Liebling, T.M., Mocellin, A., Righetti, F.:Simulating and modelling grain growthas the motion of a weighted Voronoi diagram. Mater. Sci. Forum 94–96, 301–306 (1992)

25. Uchic, M.D., Holzer, L., Inkson, B.J., Principe, E.L., Munroe, P.: Three-dimensional mi-crostructural characterization using focused ion beam tomography. MRS Bull.32, 408–416(2007)

26. Wadell, H.: Volume, shape and roundness of quartz particles. J. Geol.43(3), 250–280(1935)

27. Werz, T., Baumann, M., Wolfram, U., Krill III, C.E.: Particle tracking during Ostwald ripen-ing using time-resolved laboratory x-ray microtomography(2013). Submitted.

28. Xue, X., Righetti, F., Telley, H., Liebling, T.M.: The Laguerre model for grain growth inthree dimensions. Philos. Mag. B75(4), 567–585 (1997)